Sampath and Anjana Journal of Uncertainty Analysis and Applications (2016) 4:6 Journal of Uncertainty DOI 10.1186/s40467-016-0049-9 Analysis and Applications RESEARCH Open Access Percentile Matching Estimation of Uncertainty Distribution S Sampath* and K. Anjana * Correspondence: [email protected] Abstract Department of Statistics, University of Madras, Chennai 600005, India This paper considers the application of method of percentile matching available in statistical theory of estimation for estimating the parameters involved in uncertainty distributions. An empirical study has been carried out to compare the performance of the proposed method with the method of moments and the method of least squares considered by Wang and Peng (J. Uncertainty Analys. Appl. 2, (2014)) and Liu (Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, (2010)), respectively. The numerical study clearly establishes the superiority of the proposed method over the other two methods in estimating the parameters involved in linear uncertainty distribution when appropriate orders of percentiles are used in the estimation process. Keywords: Uncertainty distribution, Uncertain statistics, Method of least squares, Method of moments, Method of percentile matching Introduction Indeterminacy that occurs in real-life situations when the outcome of a particular event is unpredictable in advance leads to uncertainty. According to Liu [5], frequency generated from samples (historical data) and belief degree evaluated by domain experts are the two ways to explain indeterminate quantities. A fundamental premise of axiomatic approach of probability theory which came into existence in 1933 is that the estimated probability distribution should be close to the long run cumulative frequency. This ap- proach is reliable when large samples are available. In cases where samples are not available for estimating the unknown parameters of uncertainty distributions, the only choice left out is to go for belief degrees. Belief degree refers to the belief of individuals on the occur- rence of events. In order to model belief degrees, Liu [2] introduced the uncertainty theory. Since then, it has developed vigorously throughout the years. Liu [4] explains the need for uncertainty theory. Zhang [8] discusses about characteristics of uncertain measure. Liu [5] explains linear, zigzag, normal, lognormal, and empirical uncertainty distributions. Several methods are available for estimating the unknown parameters of probability distributions. Method of least squares, method of moments, and method of maximum likeli- hood are some among them. Method of moments is one of the popular methods meant for estimating parameters in a probability distribution. Method of maximum likelihood is an equally popular estimation method possessing several optimum properties. Method of least squares is a common technique mainly used for estimating parameters of regression models. © 2016 The Author(s). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Sampath and Anjana Journal of Uncertainty Analysis and Applications (2016) 4:6 Page 2 of 13 Analogous to various methods used in probability theory, estimation techniques have also been developed in uncertainty theory. Uncertain statistics refers to a methodology used for collecting and interpreting expert’s experimental data by uncertainty theory. The study of uncertain statistics was started by Liu [3]. Liu [2, 5] introduced the con- cept of moments in uncertainty theory. Wang and Peng [7] proposed the method of moments as a technique for estimating the unknown parameters of uncertainty distri- butions. Liu [5] gives detailed explanation of method of least squares, method of mo- ments, and Delphi method. Apart from these methods, exploration on the applications of alternative methods remains unattended. Method of percentile matching is an esti- mation technique used in statistical theory of estimation which plays a vital role in dealing with estimation of parameters when other popular methods fail to be effective. More details about the method of percentile matching can be found in Klugman et al. [1]. The absence of concepts like uncertainty density function makes the task of defin- ing a function similar to likelihood function (available in statistical theory) a difficult one. Hence, adopting a method similar to the maximum likelihood estimation in the uncertainty framework becomes difficult. In this paper, it is proposed to investigate the utility of the method of percentile matching in estimating the unknown parameters of uncertainty distributions. It is proposed to compare the percentile matching method with the existing competitors by way of numerical studies. The paper is organized as follows. The second section of this paper gives a detailed description on preliminary concepts of uncertainty theory. The third section deals with the commonly used estimation methods in probability theory and also explains the method of percentile matching. The fourth section is devoted for discussion on methods meant for estimating unknown parameters of uncertainty distributions. The fifth section discusses the experimental studies carried out for estimating the unknown parameters of linear uncertainty distribution. Findings and conclusions are given in the sixth section. Uncertainty Theory This section gives a short description on different terminologies associated with uncer- tainty theory due to Liu [2, 3, 5]. Certain results due to Sheng and Kar [6] on linear un- certainty distribution and details about the method developed by Wang and Peng [7] for estimating parameters of uncertainty distributions are also discussed. Let Γ be a nonempty set and ℒ be a σ-algebra over Γ. Each element Λ ∈ ℒ is called an event. A number ℳ(Λ) indicates the level that Λ will occur. Uncertain measure: Liu [2] defines a set function ℳ to be an uncertain measure if it satisfies the following three axioms: Axiom 1 (normality axiom) ℳ{Γ}=1. c Axiom 2 (duality axiom) ℳ{Λ}+ℳ{Λ }=1. Axiom 3 (subadditivity axiom) For every countable sequence of events, ∞ X∞ . ℳ ∪ Λi ≤ ℳfgΛi i¼1 i¼1 Although the probability measure satisfies the first three axioms, the probability theory is not a special case of uncertainty theory because product probability measure does not satisfy the product axiom. Sampath and Anjana Journal of Uncertainty Analysis and Applications (2016) 4:6 Page 3 of 13 Axiom 4 (product axiom) Let (Γk, ℒk, ℳk) be uncertainty spaces for k =1,2,3,…. The product uncertain measure is an uncertain measure satisfying () ∞ Y ∞ ℳ Λk ¼ ∧ ℳk fgΛk k¼1 k¼1 where Λk are arbitrarily chosen events from ℒk for k = 1,2,3,…, respectively. Uncertain variable: It is defined by Liu [2] as a measurable function ξ from an uncer- tainty space (Γ, ℒ, ℳ) to the set of real numbers such that {ξ ∈ B} is an event for any Borel set B of real numbers. Uncertainty distribution: Uncertainty distribution Φ of an uncertain variable ξ is defined by Liu [2] as Φ(x)=ℳ{ξ ≤ x}, ∀ x ∈ ℜ. Liu [5] made studies on various uncertainty distributions, namely, linear, zigzag, normal, and lognormal. This work is related to linear uncertainty distribution stated below. Linear uncertainty distribution: An uncertain variable ξ is called linear if it has uncer- tainty distribution of the form 8 <> 0; if x ≤ a x−a ΦðÞ¼x ; if a ≤ x ≤ b :> b−a 1; if x ≥ b where a and b are real numbers with a<b. It is usually denoted by ℒ (a,b). Empirical uncertainty distribution (Liu [3]): Empirical uncertainty distribution based on a given experimental data is defined as 8 > 0; if x < x1 < α −α − Φ α ðÞiþ1 i ðÞx xi ; ≤ ≤ ; ≤ < nðÞ¼x > i þ if xi x xiþ1 1 i n : xiþ1−xi 1; if x > xn where x1 < x2 < … < xn and 0 ≤ a1 ≤ a2 ≤ … ≤ an ≤ 1. Regular uncertainty distribution: An uncertainty distribution Φ(x) is said to be regu- lar by Liu [3] if it is a continuous and strictly increasing function with respect to x Φ Φ ; Φ where 0 < (x) < 1 andx lim→−∞ ðÞ¼x 0 xlim→þ∞ ðÞ¼x 1 . For example, linear, zigzag, normal, and lognormal uncertainty distributions are all regular. Expected value of an uncertain variable: The expected value of an uncertain variable ξ is defined by Liu [2] as Zþ∞ Z0 E½¼ξ ℳfgξ ≥ x dx− ℳfgξ ≤ x dx 0 −∞ provided that at least one of the two integrals is finite. It has been shown by Liu [2] that Zþ∞ Z0 E½¼ξ ðÞ1−ΦðÞx dx− ΦðÞx dx: 0 −∞ Also from this expression, using integration by parts Liu [3] gets Sampath and Anjana Journal of Uncertainty Analysis and Applications (2016) 4:6 Page 4 of 13 Zþ∞ E½¼ξ xdΦðÞx : −∞ − If ξ has a regular uncertainty distribution Φ, then by substituting Φ(x)witha, x with Φ 1(α) in the previous expression and following the change of variables of integral, Liu [3] gives Z1 − E½¼ξ Φ 1ðÞα dα: 0 Moments:Ifξ is an uncertain variable and k is a positive integer, then Liu [2] gives the kth moment of ξ as E[ξk]. Let ξ be an uncertain variable with uncertainty distribution Φ. Then by Liu [5], (i) If k is an odd number, then the kth moment of ξ is defined as hi Zþ∞ Z0 k ÀÁÀÁpffiffiffi Àpffiffiffi E ξ ¼ 1−Φ k x dx− Φ k xÞdx: 0 −∞ (ii)If k is an even number, then the kth moment of ξ is defined as hi Zþ∞ k À Àpffiffiffi À pffiffiffi E ξ ¼ 1−Φ k xÞþΦ − k xÞÞdx: 0 (iii)If k is a positive integer, then the kth moment of ξ is defined as Zþ∞ k k E½ξ ¼ x d ΦðxÞ: −∞ Sheng and Kar [6] proved that, if an uncertain variable ξ has a regular uncertainty distribution Φ and k is a positive integer, then the kth moment of ξ is Z1 hi ÀÁ − k E ξk ¼ Φ 1ðÞα dα: 0 Sheng and Kar [6] derived the expressions for the first three moments of a linear un- certain variable ξ ~ ℒ (a, b).